library(forecast)
## Registered S3 method overwritten by 'quantmod':
## method from
## as.zoo.data.frame zoo
library(quantmod)
## Loading required package: xts
## Loading required package: zoo
##
## Attaching package: 'zoo'
## The following objects are masked from 'package:base':
##
## as.Date, as.Date.numeric
## Loading required package: TTR
library(tseries)
library(timeSeries)
## Loading required package: timeDate
##
## Attaching package: 'timeSeries'
## The following object is masked from 'package:zoo':
##
## time<-
library(xts)
#Variable plug-ins
stockname = 'AMZN'
startdate = '2015-01-01'
enddate = '2019-01-01'
#Pull data from Yahoo finance
stockvar = getSymbols(c(stockname),srs='yahoo',from=startdate,toenddate=enddate,auto.assign = FALSE)
## 'getSymbols' currently uses auto.assign=TRUE by default, but will
## use auto.assign=FALSE in 0.5-0. You will still be able to use
## 'loadSymbols' to automatically load data. getOption("getSymbols.env")
## and getOption("getSymbols.auto.assign") will still be checked for
## alternate defaults.
##
## This message is shown once per session and may be disabled by setting
## options("getSymbols.warning4.0"=FALSE). See ?getSymbols for details.
stockvar = na.omit(stockvar)
#Chart time series
chartSeries(stockvar) #Represents high and low dollar value fluctuation in weekly close prices
price = stockvar[,3]
#Decompose data [Foundation of building ARIMA]
stockvar.ts = ts(price, start = 2019-01-02, frequency = 100)
stockvar.de = decompose(stockvar.ts)
plot(stockvar.de)
#Smooth data
par(mfrow = c(4,2))
#Logarithmic returns
logprice = log(price)
#Squreroot returns
sqrtprice = sqrt(price)
#Diff. log returns
dlogprice = diff(log(price),lag = 1)
dlogprice = dlogprice[!is.na(dlogprice)]
#Diff. sqrt price returns
dsqrtprice = diff(sqrt(price),lag=1)
dsqrtprice = dsqrtprice[!is.na(dsqrtprice)]
adf.test(logprice)
##
## Augmented Dickey-Fuller Test
##
## data: logprice
## Dickey-Fuller = -3.5498, Lag order = 11, p-value = 0.03751
## alternative hypothesis: stationary
adf.test(sqrtprice)
##
## Augmented Dickey-Fuller Test
##
## data: sqrtprice
## Dickey-Fuller = -2.6196, Lag order = 11, p-value = 0.316
## alternative hypothesis: stationary
adf.test(dsqrtprice)
## Warning in adf.test(dsqrtprice): p-value smaller than printed p-value
##
## Augmented Dickey-Fuller Test
##
## data: dsqrtprice
## Dickey-Fuller = -12.193, Lag order = 11, p-value = 0.01
## alternative hypothesis: stationary
adf.test(dlogprice)
## Warning in adf.test(dlogprice): p-value smaller than printed p-value
##
## Augmented Dickey-Fuller Test
##
## data: dlogprice
## Dickey-Fuller = -12.364, Lag order = 11, p-value = 0.01
## alternative hypothesis: stationary
#Creating correlograms
par(mfrow=c(1,2))
acf(dlogprice,main='ACF for log return')
pacf(dlogprice, main='PACF for log return')
par(mfrow=c(1,2))
acf(dsqrtprice,main='ACF for log return')
pacf(dsqrtprice, main='PACF for log return')
#Programming a fitted forecast
realreturn = xts(0,as.Date("2018-11-25","%Y-%m-%d"))
#Initialize forecasted return via dataframe
forecastreturn = data.frame(Forecasted = numeric())
split = floor(nrow(dlogprice)*(2.9/3))
for (s in split:(nrow(dlogprice)-1)) {
dlogprice_training = dlogprice[1:s,]
dlogprice_testing = dlogprice[(s+1):nrow(dlogprice),]
fit = arima(dlogprice_training,order = c(2,0,2),include.mean = FALSE)
summary(fit)
arima.forecast = forecast(fit, h=1)
summary(arima.forecast)
Box.test(fit$residuals,lag=1,type = 'Ljung-Box')
forecastreturn = rbind(forecastreturn,arima.forecast$mean[1])
colnames(forecastreturn) = c("Forecasted")
returnseries = dlogprice[(s+1),]
realreturn = c(realreturn,xts(returnseries))
rm(returnseries)
}
##
## Call:
## arima(x = dlogprice_training, order = c(2, 0, 2), include.mean = FALSE)
##
## Coefficients:
## ar1 ar2 ma1 ma2
## 0.3367 -0.3112 -0.2346 0.2449
## s.e. 0.3372 0.2536 0.3403 0.2718
##
## sigma^2 estimated as 0.000343: log likelihood = 3962.88, aic = -7915.76
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.001480605 0.01851997 0.01272309 99.78973 145.0924 0.7332017
## ACF1
## Training set -0.005850959
##
## Forecast method: ARIMA(2,0,2) with zero mean
##
## Model Information:
##
## Call:
## arima(x = dlogprice_training, order = c(2, 0, 2), include.mean = FALSE)
##
## Coefficients:
## ar1 ar2 ma1 ma2
## 0.3367 -0.3112 -0.2346 0.2449
## s.e. 0.3372 0.2536 0.3403 0.2718
##
## sigma^2 estimated as 0.000343: log likelihood = 3962.88, aic = -7915.76
##
## Error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.001480605 0.01851997 0.01272309 99.78973 145.0924 0.7332017
## ACF1
## Training set -0.005850959
##
## Forecasts:
## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## 1543 0.0001080032 -0.02362629 0.02384229 -0.03619046 0.03640647
##
## Call:
## arima(x = dlogprice_training, order = c(2, 0, 2), include.mean = FALSE)
##
## Coefficients:
## ar1 ar2 ma1 ma2
## 0.2014 -0.3078 -0.0970 0.2556
## s.e. 0.4320 0.2306 0.4364 0.2359
##
## sigma^2 estimated as 0.0003428: log likelihood = 3965.77, aic = -7921.53
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.001459322 0.01851619 0.01272061 100.5124 145.0138 0.7331762
## ACF1
## Training set -0.008088939
##
## Forecast method: ARIMA(2,0,2) with zero mean
##
## Model Information:
##
## Call:
## arima(x = dlogprice_training, order = c(2, 0, 2), include.mean = FALSE)
##
## Coefficients:
## ar1 ar2 ma1 ma2
## 0.2014 -0.3078 -0.0970 0.2556
## s.e. 0.4320 0.2306 0.4364 0.2359
##
## sigma^2 estimated as 0.0003428: log likelihood = 3965.77, aic = -7921.53
##
## Error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.001459322 0.01851619 0.01272061 100.5124 145.0138 0.7331762
## ACF1
## Training set -0.008088939
##
## Forecasts:
## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## 1544 -0.00109807 -0.02482752 0.02263138 -0.03738913 0.03519299
##
## Call:
## arima(x = dlogprice_training, order = c(2, 0, 2), include.mean = FALSE)
##
## Coefficients:
## ar1 ar2 ma1 ma2
## 0.2017 -0.3056 -0.0969 0.2532
## s.e. 0.4364 0.2283 0.4408 0.2338
##
## sigma^2 estimated as 0.0003429: log likelihood = 3968.14, aic = -7926.28
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.001443921 0.01851851 0.01272662 100.506 145.0733 0.7336079
## ACF1
## Training set -0.00803751
##
## Forecast method: ARIMA(2,0,2) with zero mean
##
## Model Information:
##
## Call:
## arima(x = dlogprice_training, order = c(2, 0, 2), include.mean = FALSE)
##
## Coefficients:
## ar1 ar2 ma1 ma2
## 0.2017 -0.3056 -0.0969 0.2532
## s.e. 0.4364 0.2283 0.4408 0.2338
##
## sigma^2 estimated as 0.0003429: log likelihood = 3968.14, aic = -7926.28
##
## Error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.001443921 0.01851851 0.01272662 100.506 145.0733 0.7336079
## ACF1
## Training set -0.00803751
##
## Forecasts:
## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## 1545 -0.002095314 -0.02582774 0.02163711 -0.03839092 0.03420029
##
## Call:
## arima(x = dlogprice_training, order = c(2, 0, 2), include.mean = FALSE)
##
## Coefficients:
## ar1 ar2 ma1 ma2
## 0.3410 -0.3105 -0.2377 0.2437
## s.e. 0.3386 0.2487 0.3417 0.2670
##
## sigma^2 estimated as 0.000343: log likelihood = 3970.51, aic = -7931.03
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.001441772 0.01852087 0.01273372 99.88799 145.6958 0.7344323
## ACF1
## Training set -0.005598235
##
## Forecast method: ARIMA(2,0,2) with zero mean
##
## Model Information:
##
## Call:
## arima(x = dlogprice_training, order = c(2, 0, 2), include.mean = FALSE)
##
## Coefficients:
## ar1 ar2 ma1 ma2
## 0.3410 -0.3105 -0.2377 0.2437
## s.e. 0.3386 0.2487 0.3417 0.2670
##
## sigma^2 estimated as 0.000343: log likelihood = 3970.51, aic = -7931.03
##
## Error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.001441772 0.01852087 0.01273372 99.88799 145.6958 0.7344323
## ACF1
## Training set -0.005598235
##
## Forecasts:
## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## 1546 -0.001280204 -0.02501566 0.02245525 -0.03758045 0.03502004
##
## Call:
## arima(x = dlogprice_training, order = c(2, 0, 2), include.mean = FALSE)
##
## Coefficients:
## ar1 ar2 ma1 ma2
## 0.2035 -0.3061 -0.0983 0.2533
## s.e. 0.4346 0.2263 0.4389 0.2324
##
## sigma^2 estimated as 0.0003429: log likelihood = 3973.32, aic = -7936.63
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.001434434 0.01851811 0.01273274 100.5084 145.2633 0.7338826
## ACF1
## Training set -0.007879301
##
## Forecast method: ARIMA(2,0,2) with zero mean
##
## Model Information:
##
## Call:
## arima(x = dlogprice_training, order = c(2, 0, 2), include.mean = FALSE)
##
## Coefficients:
## ar1 ar2 ma1 ma2
## 0.2035 -0.3061 -0.0983 0.2533
## s.e. 0.4346 0.2263 0.4389 0.2324
##
## sigma^2 estimated as 0.0003429: log likelihood = 3973.32, aic = -7936.63
##
## Error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.001434434 0.01851811 0.01273274 100.5084 145.2633 0.7338826
## ACF1
## Training set -0.007879301
##
## Forecasts:
## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## 1547 0.002801507 -0.0209304 0.02653341 -0.03349331 0.03909633
##
## Call:
## arima(x = dlogprice_training, order = c(2, 0, 2), include.mean = FALSE)
##
## Coefficients:
## ar1 ar2 ma1 ma2
## 0.3350 -0.3032 -0.2329 0.2378
## s.e. 0.3438 0.2516 0.3470 0.2691
##
## sigma^2 estimated as 0.0003432: log likelihood = 3975.31, aic = -7940.63
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.001428838 0.01852495 0.01274228 99.98143 144.9518 0.7339396
## ACF1
## Training set -0.00547146
##
## Forecast method: ARIMA(2,0,2) with zero mean
##
## Model Information:
##
## Call:
## arima(x = dlogprice_training, order = c(2, 0, 2), include.mean = FALSE)
##
## Coefficients:
## ar1 ar2 ma1 ma2
## 0.3350 -0.3032 -0.2329 0.2378
## s.e. 0.3438 0.2516 0.3470 0.2691
##
## sigma^2 estimated as 0.0003432: log likelihood = 3975.31, aic = -7940.63
##
## Error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.001428838 0.01852495 0.01274228 99.98143 144.9518 0.7339396
## ACF1
## Training set -0.00547146
##
## Forecasts:
## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## 1548 -0.002263543 -0.02600422 0.02147713 -0.03857177 0.03404468
##
## Call:
## arima(x = dlogprice_training, order = c(2, 0, 2), include.mean = FALSE)
##
## Coefficients:
## ar1 ar2 ma1 ma2
## 0.3340 -0.3029 -0.2317 0.2375
## s.e. 0.3437 0.2512 0.3468 0.2686
##
## sigma^2 estimated as 0.000343: log likelihood = 3978.38, aic = -7946.76
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.001426886 0.01851899 0.01273495 99.94292 144.8959 0.7334036
## ACF1
## Training set -0.005388152
##
## Forecast method: ARIMA(2,0,2) with zero mean
##
## Model Information:
##
## Call:
## arima(x = dlogprice_training, order = c(2, 0, 2), include.mean = FALSE)
##
## Coefficients:
## ar1 ar2 ma1 ma2
## 0.3340 -0.3029 -0.2317 0.2375
## s.e. 0.3437 0.2512 0.3468 0.2686
##
## sigma^2 estimated as 0.000343: log likelihood = 3978.38, aic = -7946.76
##
## Error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.001426886 0.01851899 0.01273495 99.94292 144.8959 0.7334036
## ACF1
## Training set -0.005388152
##
## Forecasts:
## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## 1549 6.354664e-05 -0.0236695 0.02379659 -0.03623302 0.03636011
##
## Call:
## arima(x = dlogprice_training, order = c(2, 0, 2), include.mean = FALSE)
##
## Coefficients:
## ar1 ar2 ma1 ma2
## 0.3436 -0.2917 -0.2414 0.2243
## s.e. 0.3526 0.2516 0.3555 0.2704
##
## sigma^2 estimated as 0.000343: log likelihood = 3980.87, aic = -7951.75
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.001440528 0.01851991 0.01273954 99.84719 145.0927 0.7334977
## ACF1
## Training set -0.005509256
##
## Forecast method: ARIMA(2,0,2) with zero mean
##
## Model Information:
##
## Call:
## arima(x = dlogprice_training, order = c(2, 0, 2), include.mean = FALSE)
##
## Coefficients:
## ar1 ar2 ma1 ma2
## 0.3436 -0.2917 -0.2414 0.2243
## s.e. 0.3526 0.2516 0.3555 0.2704
##
## sigma^2 estimated as 0.000343: log likelihood = 3980.87, aic = -7951.75
##
## Error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.001440528 0.01851991 0.01273954 99.84719 145.0927 0.7334977
## ACF1
## Training set -0.005509256
##
## Forecasts:
## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## 1550 0.002829266 -0.02090496 0.02656349 -0.0334691 0.03912763
##
## Call:
## arima(x = dlogprice_training, order = c(2, 0, 2), include.mean = FALSE)
##
## Coefficients:
## ar1 ar2 ma1 ma2
## 0.3443 -0.2907 -0.2422 0.2233
## s.e. 0.3519 0.2534 0.3548 0.2720
##
## sigma^2 estimated as 0.0003428: log likelihood = 3983.89, aic = -7957.77
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.001435777 0.01851464 0.01273531 99.8913 145.0173 0.7330872
## ACF1
## Training set -0.005554807
##
## Forecast method: ARIMA(2,0,2) with zero mean
##
## Model Information:
##
## Call:
## arima(x = dlogprice_training, order = c(2, 0, 2), include.mean = FALSE)
##
## Coefficients:
## ar1 ar2 ma1 ma2
## 0.3443 -0.2907 -0.2422 0.2233
## s.e. 0.3519 0.2534 0.3548 0.2720
##
## sigma^2 estimated as 0.0003428: log likelihood = 3983.89, aic = -7957.77
##
## Error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.001435777 0.01851464 0.01273531 99.8913 145.0173 0.7330872
## ACF1
## Training set -0.005554807
##
## Forecasts:
## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## 1551 -0.001050912 -0.02477838 0.02267655 -0.03733894 0.03523711
##
## Call:
## arima(x = dlogprice_training, order = c(2, 0, 2), include.mean = FALSE)
##
## Coefficients:
## ar1 ar2 ma1 ma2
## 0.3374 -0.2808 -0.2349 0.2126
## s.e. 0.3564 0.2447 0.3595 0.2622
##
## sigma^2 estimated as 0.0003431: log likelihood = 3985.71, aic = -7961.42
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.001416291 0.01852353 0.01274574 99.74808 145.0895 0.733431
## ACF1
## Training set -0.005585046
##
## Forecast method: ARIMA(2,0,2) with zero mean
##
## Model Information:
##
## Call:
## arima(x = dlogprice_training, order = c(2, 0, 2), include.mean = FALSE)
##
## Coefficients:
## ar1 ar2 ma1 ma2
## 0.3374 -0.2808 -0.2349 0.2126
## s.e. 0.3564 0.2447 0.3595 0.2622
##
## sigma^2 estimated as 0.0003431: log likelihood = 3985.71, aic = -7961.42
##
## Error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.001416291 0.01852353 0.01274574 99.74808 145.0895 0.733431
## ACF1
## Training set -0.005585046
##
## Forecasts:
## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## 1552 -0.003725869 -0.02746473 0.02001299 -0.04003132 0.03257958
##
## Call:
## arima(x = dlogprice_training, order = c(2, 0, 2), include.mean = FALSE)
##
## Coefficients:
## ar1 ar2 ma1 ma2
## 0.3385 -0.2861 -0.2354 0.2178
## s.e. 0.3517 0.2427 0.3548 0.2602
##
## sigma^2 estimated as 0.000343: log likelihood = 3988.54, aic = -7967.07
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.001406486 0.01852048 0.01274625 99.76525 145.3913 0.7335631
## ACF1
## Training set -0.005390457
##
## Forecast method: ARIMA(2,0,2) with zero mean
##
## Model Information:
##
## Call:
## arima(x = dlogprice_training, order = c(2, 0, 2), include.mean = FALSE)
##
## Coefficients:
## ar1 ar2 ma1 ma2
## 0.3385 -0.2861 -0.2354 0.2178
## s.e. 0.3517 0.2427 0.3548 0.2602
##
## sigma^2 estimated as 0.000343: log likelihood = 3988.54, aic = -7967.07
##
## Error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.001406486 0.01852048 0.01274625 99.76525 145.3913 0.7335631
## ACF1
## Training set -0.005390457
##
## Forecasts:
## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## 1553 -0.0003001465 -0.02403509 0.0234348 -0.03659961 0.03599932
##
## Call:
## arima(x = dlogprice_training, order = c(2, 0, 2), include.mean = FALSE)
##
## Coefficients:
## ar1 ar2 ma1 ma2
## 0.3439 -0.2892 -0.2403 0.2216
## s.e. 0.3520 0.2456 0.3550 0.2635
##
## sigma^2 estimated as 0.0003431: log likelihood = 3990.92, aic = -7971.83
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.001389718 0.01852277 0.01275262 100.0076 145.6516 0.7342549
## ACF1
## Training set -0.00518322
##
## Forecast method: ARIMA(2,0,2) with zero mean
##
## Model Information:
##
## Call:
## arima(x = dlogprice_training, order = c(2, 0, 2), include.mean = FALSE)
##
## Coefficients:
## ar1 ar2 ma1 ma2
## 0.3439 -0.2892 -0.2403 0.2216
## s.e. 0.3520 0.2456 0.3550 0.2635
##
## sigma^2 estimated as 0.0003431: log likelihood = 3990.92, aic = -7971.83
##
## Error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.001389718 0.01852277 0.01275262 100.0076 145.6516 0.7342549
## ACF1
## Training set -0.00518322
##
## Forecasts:
## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## 1554 -0.0004101203 -0.024148 0.02332776 -0.03671408 0.03589384
##
## Call:
## arima(x = dlogprice_training, order = c(2, 0, 2), include.mean = FALSE)
##
## Coefficients:
## ar1 ar2 ma1 ma2
## 0.3418 -0.3110 -0.2394 0.2436
## s.e. 0.3330 0.2475 0.3362 0.2645
##
## sigma^2 estimated as 0.0003433: log likelihood = 3993.11, aic = -7976.22
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.00140719 0.01852722 0.01275959 99.72719 145.0486 0.733871
## ACF1
## Training set -0.00502779
##
## Forecast method: ARIMA(2,0,2) with zero mean
##
## Model Information:
##
## Call:
## arima(x = dlogprice_training, order = c(2, 0, 2), include.mean = FALSE)
##
## Coefficients:
## ar1 ar2 ma1 ma2
## 0.3418 -0.3110 -0.2394 0.2436
## s.e. 0.3330 0.2475 0.3362 0.2645
##
## sigma^2 estimated as 0.0003433: log likelihood = 3993.11, aic = -7976.22
##
## Error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.00140719 0.01852722 0.01275959 99.72719 145.0486 0.733871
## ACF1
## Training set -0.00502779
##
## Forecasts:
## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## 1555 0.003969307 -0.01977428 0.02771289 -0.03234337 0.04028199
##
## Call:
## arima(x = dlogprice_training, order = c(2, 0, 2), include.mean = FALSE)
##
## Coefficients:
## ar1 ar2 ma1 ma2
## 0.3484 -0.2938 -0.2448 0.2245
## s.e. 0.3400 0.2451 0.3433 0.2628
##
## sigma^2 estimated as 0.0003432: log likelihood = 3995.89, aic = -7981.78
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.001415228 0.01852471 0.01276088 99.74786 145.7019 0.7342541
## ACF1
## Training set -0.005595217
##
## Forecast method: ARIMA(2,0,2) with zero mean
##
## Model Information:
##
## Call:
## arima(x = dlogprice_training, order = c(2, 0, 2), include.mean = FALSE)
##
## Coefficients:
## ar1 ar2 ma1 ma2
## 0.3484 -0.2938 -0.2448 0.2245
## s.e. 0.3400 0.2451 0.3433 0.2628
##
## sigma^2 estimated as 0.0003432: log likelihood = 3995.89, aic = -7981.78
##
## Error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.001415228 0.01852471 0.01276088 99.74786 145.7019 0.7342541
## ACF1
## Training set -0.005595217
##
## Forecasts:
## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## 1556 0.001264959 -0.02247541 0.02500533 -0.03504281 0.03757273
##
## Call:
## arima(x = dlogprice_training, order = c(2, 0, 2), include.mean = FALSE)
##
## Coefficients:
## ar1 ar2 ma1 ma2
## 0.3510 -0.2934 -0.2473 0.2241
## s.e. 0.3422 0.2439 0.3454 0.2619
##
## sigma^2 estimated as 0.000343: log likelihood = 3998.89, aic = -7987.78
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.001418382 0.0185196 0.01275735 99.80198 145.8228 0.7342561
## ACF1
## Training set -0.005690426
##
## Forecast method: ARIMA(2,0,2) with zero mean
##
## Model Information:
##
## Call:
## arima(x = dlogprice_training, order = c(2, 0, 2), include.mean = FALSE)
##
## Coefficients:
## ar1 ar2 ma1 ma2
## 0.3510 -0.2934 -0.2473 0.2241
## s.e. 0.3422 0.2439 0.3454 0.2619
##
## sigma^2 estimated as 0.000343: log likelihood = 3998.89, aic = -7987.78
##
## Error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.001418382 0.0185196 0.01275735 99.80198 145.8228 0.7342561
## ACF1
## Training set -0.005690426
##
## Forecasts:
## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## 1557 -0.0009802688 -0.02471409 0.02275356 -0.03727802 0.03531748
##
## Call:
## arima(x = dlogprice_training, order = c(2, 0, 2), include.mean = FALSE)
##
## Coefficients:
## ar1 ar2 ma1 ma2
## 0.3444 -0.2799 -0.2401 0.2114
## s.e. 0.3607 0.2427 0.3638 0.2616
##
## sigma^2 estimated as 0.000343: log likelihood = 4001.47, aic = -7992.94
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.001426944 0.01851947 0.01276098 99.96821 145.8313 0.734691
## ACF1
## Training set -0.005982787
##
## Forecast method: ARIMA(2,0,2) with zero mean
##
## Model Information:
##
## Call:
## arima(x = dlogprice_training, order = c(2, 0, 2), include.mean = FALSE)
##
## Coefficients:
## ar1 ar2 ma1 ma2
## 0.3444 -0.2799 -0.2401 0.2114
## s.e. 0.3607 0.2427 0.3638 0.2616
##
## sigma^2 estimated as 0.000343: log likelihood = 4001.47, aic = -7992.94
##
## Error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.001426944 0.01851947 0.01276098 99.96821 145.8313 0.734691
## ACF1
## Training set -0.005982787
##
## Forecasts:
## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## 1558 0.0007475144 -0.02298614 0.02448117 -0.03554998 0.03704501
##
## Call:
## arima(x = dlogprice_training, order = c(2, 0, 2), include.mean = FALSE)
##
## Coefficients:
## ar1 ar2 ma1 ma2
## 0.358 -0.2863 -0.2544 0.2166
## s.e. 0.351 0.2448 0.3541 0.2638
##
## sigma^2 estimated as 0.0003429: log likelihood = 4004.3, aic = -7998.6
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.001419997 0.01851639 0.01276108 99.90276 145.8453 0.7343671
## ACF1
## Training set -0.005864622
##
## Forecast method: ARIMA(2,0,2) with zero mean
##
## Model Information:
##
## Call:
## arima(x = dlogprice_training, order = c(2, 0, 2), include.mean = FALSE)
##
## Coefficients:
## ar1 ar2 ma1 ma2
## 0.358 -0.2863 -0.2544 0.2166
## s.e. 0.351 0.2448 0.3541 0.2638
##
## sigma^2 estimated as 0.0003429: log likelihood = 4004.3, aic = -7998.6
##
## Error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.001419997 0.01851639 0.01276108 99.90276 145.8453 0.7343671
## ACF1
## Training set -0.005864622
##
## Forecasts:
## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## 1559 -0.002091077 -0.02582079 0.02163864 -0.03838254 0.03420039
##
## Call:
## arima(x = dlogprice_training, order = c(2, 0, 2), include.mean = FALSE)
##
## Coefficients:
## ar1 ar2 ma1 ma2
## 0.3592 -0.2850 -0.2554 0.215
## s.e. 0.3515 0.2441 0.3546 0.263
##
## sigma^2 estimated as 0.0003426: log likelihood = 4007.36, aic = -8004.73
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.001417701 0.01851055 0.01275441 99.87006 145.844 0.7342428
## ACF1
## Training set -0.005836903
##
## Forecast method: ARIMA(2,0,2) with zero mean
##
## Model Information:
##
## Call:
## arima(x = dlogprice_training, order = c(2, 0, 2), include.mean = FALSE)
##
## Coefficients:
## ar1 ar2 ma1 ma2
## 0.3592 -0.2850 -0.2554 0.215
## s.e. 0.3515 0.2441 0.3546 0.263
##
## sigma^2 estimated as 0.0003426: log likelihood = 4007.36, aic = -8004.73
##
## Error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.001417701 0.01851055 0.01275441 99.87006 145.844 0.7342428
## ACF1
## Training set -0.005836903
##
## Forecasts:
## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## 1560 -0.0002890042 -0.02401123 0.02343322 -0.03656901 0.035991
##
## Call:
## arima(x = dlogprice_training, order = c(2, 0, 2), include.mean = FALSE)
##
## Coefficients:
## ar1 ar2 ma1 ma2
## 0.3568 -0.2835 -0.2532 0.2135
## s.e. 0.3464 0.2452 0.3496 0.2632
##
## sigma^2 estimated as 0.0003426: log likelihood = 4010.12, aic = -8010.24
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.001426463 0.01850832 0.01275546 99.81778 145.7173 0.7342662
## ACF1
## Training set -0.005820756
##
## Forecast method: ARIMA(2,0,2) with zero mean
##
## Model Information:
##
## Call:
## arima(x = dlogprice_training, order = c(2, 0, 2), include.mean = FALSE)
##
## Coefficients:
## ar1 ar2 ma1 ma2
## 0.3568 -0.2835 -0.2532 0.2135
## s.e. 0.3464 0.2452 0.3496 0.2632
##
## sigma^2 estimated as 0.0003426: log likelihood = 4010.12, aic = -8010.24
##
## Error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.001426463 0.01850832 0.01275546 99.81778 145.7173 0.7342662
## ACF1
## Training set -0.005820756
##
## Forecasts:
## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## 1561 0.002170725 -0.02154865 0.02589009 -0.03410492 0.03844637
##
## Call:
## arima(x = dlogprice_training, order = c(2, 0, 2), include.mean = FALSE)
##
## Coefficients:
## ar1 ar2 ma1 ma2
## 0.3568 -0.2831 -0.2533 0.2131
## s.e. 0.3476 0.2459 0.3507 0.2642
##
## sigma^2 estimated as 0.0003423: log likelihood = 4013.17, aic = -8016.34
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.001423014 0.01850267 0.01274982 99.89774 145.7068 0.7339737
## ACF1
## Training set -0.005844969
##
## Forecast method: ARIMA(2,0,2) with zero mean
##
## Model Information:
##
## Call:
## arima(x = dlogprice_training, order = c(2, 0, 2), include.mean = FALSE)
##
## Coefficients:
## ar1 ar2 ma1 ma2
## 0.3568 -0.2831 -0.2533 0.2131
## s.e. 0.3476 0.2459 0.3507 0.2642
##
## sigma^2 estimated as 0.0003423: log likelihood = 4013.17, aic = -8016.34
##
## Error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.001423014 0.01850267 0.01274982 99.89774 145.7068 0.7339737
## ACF1
## Training set -0.005844969
##
## Forecasts:
## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## 1562 -0.000583117 -0.02429525 0.02312901 -0.03684769 0.03568145
##
## Call:
## arima(x = dlogprice_training, order = c(2, 0, 2), include.mean = FALSE)
##
## Coefficients:
## ar1 ar2 ma1 ma2
## 0.3502 -0.2796 -0.2465 0.2098
## s.e. 0.3500 0.2452 0.3531 0.2634
##
## sigma^2 estimated as 0.0003423: log likelihood = 4015.94, aic = -8021.88
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.001412672 0.01850028 0.01275066 99.84133 145.6333 0.7341405
## ACF1
## Training set -0.00588337
##
## Forecast method: ARIMA(2,0,2) with zero mean
##
## Model Information:
##
## Call:
## arima(x = dlogprice_training, order = c(2, 0, 2), include.mean = FALSE)
##
## Coefficients:
## ar1 ar2 ma1 ma2
## 0.3502 -0.2796 -0.2465 0.2098
## s.e. 0.3500 0.2452 0.3531 0.2634
##
## sigma^2 estimated as 0.0003423: log likelihood = 4015.94, aic = -8021.88
##
## Error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.001412672 0.01850028 0.01275066 99.84133 145.6333 0.7341405
## ACF1
## Training set -0.00588337
##
## Forecasts:
## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## 1563 -0.002013532 -0.02572259 0.02169553 -0.03827341 0.03424635
##
## Call:
## arima(x = dlogprice_training, order = c(2, 0, 2), include.mean = FALSE)
##
## Coefficients:
## ar1 ar2 ma1 ma2
## 0.3510 -0.2797 -0.2473 0.2098
## s.e. 0.3494 0.2451 0.3525 0.2633
##
## sigma^2 estimated as 0.000342: log likelihood = 4019.01, aic = -8028.02
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.001411342 0.01849437 0.012743 99.7971 145.5764 0.7338431
## ACF1
## Training set -0.005823643
##
## Forecast method: ARIMA(2,0,2) with zero mean
##
## Model Information:
##
## Call:
## arima(x = dlogprice_training, order = c(2, 0, 2), include.mean = FALSE)
##
## Coefficients:
## ar1 ar2 ma1 ma2
## 0.3510 -0.2797 -0.2473 0.2098
## s.e. 0.3494 0.2451 0.3525 0.2633
##
## sigma^2 estimated as 0.000342: log likelihood = 4019.01, aic = -8028.02
##
## Error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.001411342 0.01849437 0.012743 99.7971 145.5764 0.7338431
## ACF1
## Training set -0.005823643
##
## Forecasts:
## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## 1564 0.0003770919 -0.02332439 0.02407858 -0.0358712 0.03662539
##
## Call:
## arima(x = dlogprice_training, order = c(2, 0, 2), include.mean = FALSE)
##
## Coefficients:
## ar1 ar2 ma1 ma2
## 0.3486 -0.2777 -0.2449 0.2076
## s.e. 0.3508 0.2435 0.3540 0.2618
##
## sigma^2 estimated as 0.0003419: log likelihood = 4021.8, aic = -8033.6
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.001419652 0.01849181 0.01274363 99.73078 145.502 0.7338873
## ACF1
## Training set -0.005874706
##
## Forecast method: ARIMA(2,0,2) with zero mean
##
## Model Information:
##
## Call:
## arima(x = dlogprice_training, order = c(2, 0, 2), include.mean = FALSE)
##
## Coefficients:
## ar1 ar2 ma1 ma2
## 0.3486 -0.2777 -0.2449 0.2076
## s.e. 0.3508 0.2435 0.3540 0.2618
##
## sigma^2 estimated as 0.0003419: log likelihood = 4021.8, aic = -8033.6
##
## Error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.001419652 0.01849181 0.01274363 99.73078 145.502 0.7338873
## ACF1
## Training set -0.005874706
##
## Forecasts:
## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## 1565 0.002188571 -0.02150964 0.02588678 -0.03405471 0.03843185
##
## Call:
## arima(x = dlogprice_training, order = c(2, 0, 2), include.mean = FALSE)
##
## Coefficients:
## ar1 ar2 ma1 ma2
## 0.3501 -0.2860 -0.2461 0.2159
## s.e. 0.3440 0.2439 0.3471 0.2623
##
## sigma^2 estimated as 0.0003419: log likelihood = 4024.43, aic = -8038.85
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.001429831 0.01849115 0.01274689 99.72377 145.6898 0.7343989
## ACF1
## Training set -0.005843077
##
## Forecast method: ARIMA(2,0,2) with zero mean
##
## Model Information:
##
## Call:
## arima(x = dlogprice_training, order = c(2, 0, 2), include.mean = FALSE)
##
## Coefficients:
## ar1 ar2 ma1 ma2
## 0.3501 -0.2860 -0.2461 0.2159
## s.e. 0.3440 0.2439 0.3471 0.2623
##
## sigma^2 estimated as 0.0003419: log likelihood = 4024.43, aic = -8038.85
##
## Error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.001429831 0.01849115 0.01274689 99.72377 145.6898 0.7343989
## ACF1
## Training set -0.005843077
##
## Forecasts:
## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## 1566 0.001508138 -0.02218923 0.0252055 -0.03473385 0.03775013
##
## Call:
## arima(x = dlogprice_training, order = c(2, 0, 2), include.mean = FALSE)
##
## Coefficients:
## ar1 ar2 ma1 ma2
## 0.3464 -0.2856 -0.2427 0.2155
## s.e. 0.3447 0.2433 0.3479 0.2616
##
## sigma^2 estimated as 0.0003418: log likelihood = 4027.25, aic = -8044.5
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.001421241 0.01848817 0.01274676 99.64546 145.4055 0.7340179
## ACF1
## Training set -0.005874378
##
## Forecast method: ARIMA(2,0,2) with zero mean
##
## Model Information:
##
## Call:
## arima(x = dlogprice_training, order = c(2, 0, 2), include.mean = FALSE)
##
## Coefficients:
## ar1 ar2 ma1 ma2
## 0.3464 -0.2856 -0.2427 0.2155
## s.e. 0.3447 0.2433 0.3479 0.2616
##
## sigma^2 estimated as 0.0003418: log likelihood = 4027.25, aic = -8044.5
##
## Error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.001421241 0.01848817 0.01274676 99.64546 145.4055 0.7340179
## ACF1
## Training set -0.005874378
##
## Forecasts:
## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## 1567 -0.002684923 -0.02637847 0.02100862 -0.03892107 0.03355123
##
## Call:
## arima(x = dlogprice_training, order = c(2, 0, 2), include.mean = FALSE)
##
## Coefficients:
## ar1 ar2 ma1 ma2
## 0.3490 -0.2905 -0.2451 0.2198
## s.e. 0.3378 0.2437 0.3410 0.2621
##
## sigma^2 estimated as 0.0003417: log likelihood = 4030.08, aic = -8050.15
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.001412798 0.01848517 0.01274699 99.55817 145.5523 0.7343873
## ACF1
## Training set -0.005701744
##
## Forecast method: ARIMA(2,0,2) with zero mean
##
## Model Information:
##
## Call:
## arima(x = dlogprice_training, order = c(2, 0, 2), include.mean = FALSE)
##
## Coefficients:
## ar1 ar2 ma1 ma2
## 0.3490 -0.2905 -0.2451 0.2198
## s.e. 0.3378 0.2437 0.3410 0.2621
##
## sigma^2 estimated as 0.0003417: log likelihood = 4030.08, aic = -8050.15
##
## Error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.001412798 0.01848517 0.01274699 99.55817 145.5523 0.7343873
## ACF1
## Training set -0.005701744
##
## Forecasts:
## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## 1568 -0.001808345 -0.02549804 0.02188135 -0.03803861 0.03442192
##
## Call:
## arima(x = dlogprice_training, order = c(2, 0, 2), include.mean = FALSE)
##
## Coefficients:
## ar1 ar2 ma1 ma2
## 0.3552 -0.2957 -0.2511 0.2248
## s.e. 0.3329 0.2445 0.3360 0.2630
##
## sigma^2 estimated as 0.0003416: log likelihood = 4032.95, aic = -8055.89
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.001404433 0.01848167 0.01274664 99.59114 145.7494 0.7347811
## ACF1
## Training set -0.005545206
##
## Forecast method: ARIMA(2,0,2) with zero mean
##
## Model Information:
##
## Call:
## arima(x = dlogprice_training, order = c(2, 0, 2), include.mean = FALSE)
##
## Coefficients:
## ar1 ar2 ma1 ma2
## 0.3552 -0.2957 -0.2511 0.2248
## s.e. 0.3329 0.2445 0.3360 0.2630
##
## sigma^2 estimated as 0.0003416: log likelihood = 4032.95, aic = -8055.89
##
## Error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.001404433 0.01848167 0.01274664 99.59114 145.7494 0.7347811
## ACF1
## Training set -0.005545206
##
## Forecasts:
## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## 1569 -0.000157118 -0.02384233 0.02352809 -0.03638052 0.03606629
##
## Call:
## arima(x = dlogprice_training, order = c(2, 0, 2), include.mean = FALSE)
##
## Coefficients:
## ar1 ar2 ma1 ma2
## 0.3541 -0.3018 -0.2503 0.2309
## s.e. 0.3269 0.2453 0.3300 0.2636
##
## sigma^2 estimated as 0.0003414: log likelihood = 4035.84, aic = -8061.69
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.00141134 0.01847782 0.01274527 99.49412 145.5466 0.7345113
## ACF1
## Training set -0.005437234
##
## Forecast method: ARIMA(2,0,2) with zero mean
##
## Model Information:
##
## Call:
## arima(x = dlogprice_training, order = c(2, 0, 2), include.mean = FALSE)
##
## Coefficients:
## ar1 ar2 ma1 ma2
## 0.3541 -0.3018 -0.2503 0.2309
## s.e. 0.3269 0.2453 0.3300 0.2636
##
## sigma^2 estimated as 0.0003414: log likelihood = 4035.84, aic = -8061.69
##
## Error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.00141134 0.01847782 0.01274527 99.49412 145.5466 0.7345113
## ACF1
## Training set -0.005437234
##
## Forecasts:
## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## 1570 0.002477296 -0.02120299 0.02615758 -0.03373857 0.03869316
##
## Call:
## arima(x = dlogprice_training, order = c(2, 0, 2), include.mean = FALSE)
##
## Coefficients:
## ar1 ar2 ma1 ma2
## 0.3538 -0.3016 -0.2500 0.2307
## s.e. 0.3271 0.2451 0.3303 0.2634
##
## sigma^2 estimated as 0.0003412: log likelihood = 4038.92, aic = -8067.83
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.001409984 0.01847195 0.01273756 99.41118 145.4649 0.7342937
## ACF1
## Training set -0.005466099
##
## Forecast method: ARIMA(2,0,2) with zero mean
##
## Model Information:
##
## Call:
## arima(x = dlogprice_training, order = c(2, 0, 2), include.mean = FALSE)
##
## Coefficients:
## ar1 ar2 ma1 ma2
## 0.3538 -0.3016 -0.2500 0.2307
## s.e. 0.3271 0.2451 0.3303 0.2634
##
## sigma^2 estimated as 0.0003412: log likelihood = 4038.92, aic = -8067.83
##
## Error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.001409984 0.01847195 0.01273756 99.41118 145.4649 0.7342937
## ACF1
## Training set -0.005466099
##
## Forecasts:
## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## 1571 7.571212e-05 -0.02359704 0.02374846 -0.03612864 0.03628006
##
## Call:
## arima(x = dlogprice_training, order = c(2, 0, 2), include.mean = FALSE)
##
## Coefficients:
## ar1 ar2 ma1 ma2
## 0.3589 -0.3022 -0.2552 0.2310
## s.e. 0.3265 0.2449 0.3297 0.2634
##
## sigma^2 estimated as 0.0003411: log likelihood = 4041.86, aic = -8073.72
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.001415242 0.01846754 0.01273546 99.43807 145.5439 0.7344377
## ACF1
## Training set -0.005487437
##
## Forecast method: ARIMA(2,0,2) with zero mean
##
## Model Information:
##
## Call:
## arima(x = dlogprice_training, order = c(2, 0, 2), include.mean = FALSE)
##
## Coefficients:
## ar1 ar2 ma1 ma2
## 0.3589 -0.3022 -0.2552 0.2310
## s.e. 0.3265 0.2449 0.3297 0.2634
##
## sigma^2 estimated as 0.0003411: log likelihood = 4041.86, aic = -8073.72
##
## Error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.001415242 0.01846754 0.01273546 99.43807 145.5439 0.7344377
## ACF1
## Training set -0.005487437
##
## Forecasts:
## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## 1572 0.0002887444 -0.02337837 0.02395585 -0.03590698 0.03648447
##
## Call:
## arima(x = dlogprice_training, order = c(2, 0, 2), include.mean = FALSE)
##
## Coefficients:
## ar1 ar2 ma1 ma2
## 0.3584 -0.2911 -0.2542 0.2196
## s.e. 0.3379 0.2420 0.3410 0.2609
##
## sigma^2 estimated as 0.000341: log likelihood = 4044.52, aic = -8079.03
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.001424239 0.01846659 0.01273825 99.51764 145.7167 0.7348551
## ACF1
## Training set -0.005743757
##
## Forecast method: ARIMA(2,0,2) with zero mean
##
## Model Information:
##
## Call:
## arima(x = dlogprice_training, order = c(2, 0, 2), include.mean = FALSE)
##
## Coefficients:
## ar1 ar2 ma1 ma2
## 0.3584 -0.2911 -0.2542 0.2196
## s.e. 0.3379 0.2420 0.3410 0.2609
##
## sigma^2 estimated as 0.000341: log likelihood = 4044.52, aic = -8079.03
##
## Error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.001424239 0.01846659 0.01273825 99.51764 145.7167 0.7348551
## ACF1
## Training set -0.005743757
##
## Forecasts:
## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## 1573 0.001185614 -0.02248027 0.0248515 -0.03500823 0.03737946
##
## Call:
## arima(x = dlogprice_training, order = c(2, 0, 2), include.mean = FALSE)
##
## Coefficients:
## ar1 ar2 ma1 ma2
## 0.3522 -0.2887 -0.2475 0.2177
## s.e. 0.3432 0.2403 0.3462 0.2595
##
## sigma^2 estimated as 0.0003409: log likelihood = 4047.33, aic = -8084.65
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.001430423 0.0184638 0.01273877 99.57158 145.7743 0.7352825
## ACF1
## Training set -0.00589391
##
## Forecast method: ARIMA(2,0,2) with zero mean
##
## Model Information:
##
## Call:
## arima(x = dlogprice_training, order = c(2, 0, 2), include.mean = FALSE)
##
## Coefficients:
## ar1 ar2 ma1 ma2
## 0.3522 -0.2887 -0.2475 0.2177
## s.e. 0.3432 0.2403 0.3462 0.2595
##
## sigma^2 estimated as 0.0003409: log likelihood = 4047.33, aic = -8084.65
##
## Error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.001430423 0.0184638 0.01273877 99.57158 145.7743 0.7352825
## ACF1
## Training set -0.00589391
##
## Forecasts:
## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## 1574 0.0005314097 -0.02313091 0.02419372 -0.03565698 0.0367198
##
## Call:
## arima(x = dlogprice_training, order = c(2, 0, 2), include.mean = FALSE)
##
## Coefficients:
## ar1 ar2 ma1 ma2
## 0.3468 -0.2907 -0.2417 0.2210
## s.e. 0.3432 0.2415 0.3462 0.2608
##
## sigma^2 estimated as 0.0003409: log likelihood = 4049.98, aic = -8089.96
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.001438515 0.01846288 0.01274163 99.68675 145.7803 0.7358361
## ACF1
## Training set -0.005896909
##
## Forecast method: ARIMA(2,0,2) with zero mean
##
## Model Information:
##
## Call:
## arima(x = dlogprice_training, order = c(2, 0, 2), include.mean = FALSE)
##
## Coefficients:
## ar1 ar2 ma1 ma2
## 0.3468 -0.2907 -0.2417 0.2210
## s.e. 0.3432 0.2415 0.3462 0.2608
##
## sigma^2 estimated as 0.0003409: log likelihood = 4049.98, aic = -8089.96
##
## Error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.001438515 0.01846288 0.01274163 99.68675 145.7803 0.7358361
## ACF1
## Training set -0.005896909
##
## Forecasts:
## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## 1575 0.0006872809 -0.02297386 0.02434842 -0.03549931 0.03687387
##
## Call:
## arima(x = dlogprice_training, order = c(2, 0, 2), include.mean = FALSE)
##
## Coefficients:
## ar1 ar2 ma1 ma2
## 0.3464 -0.2904 -0.2413 0.2207
## s.e. 0.3432 0.2414 0.3462 0.2607
##
## sigma^2 estimated as 0.0003407: log likelihood = 4053.05, aic = -8096.1
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.001438326 0.01845705 0.01273442 99.67543 145.7419 0.7354704
## ACF1
## Training set -0.005942696
##
## Forecast method: ARIMA(2,0,2) with zero mean
##
## Model Information:
##
## Call:
## arima(x = dlogprice_training, order = c(2, 0, 2), include.mean = FALSE)
##
## Coefficients:
## ar1 ar2 ma1 ma2
## 0.3464 -0.2904 -0.2413 0.2207
## s.e. 0.3432 0.2414 0.3462 0.2607
##
## sigma^2 estimated as 0.0003407: log likelihood = 4053.05, aic = -8096.1
##
## Error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.001438326 0.01845705 0.01273442 99.67543 145.7419 0.7354704
## ACF1
## Training set -0.005942696
##
## Forecasts:
## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## 1576 -0.0009602613 -0.02461393 0.02269341 -0.03713542 0.0352149
##
## Call:
## arima(x = dlogprice_training, order = c(2, 0, 2), include.mean = FALSE)
##
## Coefficients:
## ar1 ar2 ma1 ma2
## 0.3365 -0.2868 -0.2312 0.2186
## s.e. 0.3469 0.2390 0.3500 0.2570
##
## sigma^2 estimated as 0.0003408: log likelihood = 4055.41, aic = -8100.83
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.001449269 0.01845948 0.01274023 99.80221 145.5329 0.7357619
## ACF1
## Training set -0.006049279
##
## Forecast method: ARIMA(2,0,2) with zero mean
##
## Model Information:
##
## Call:
## arima(x = dlogprice_training, order = c(2, 0, 2), include.mean = FALSE)
##
## Coefficients:
## ar1 ar2 ma1 ma2
## 0.3365 -0.2868 -0.2312 0.2186
## s.e. 0.3469 0.2390 0.3500 0.2570
##
## sigma^2 estimated as 0.0003408: log likelihood = 4055.41, aic = -8100.83
##
## Error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.001449269 0.01845948 0.01274023 99.80221 145.5329 0.7357619
## ACF1
## Training set -0.006049279
##
## Forecasts:
## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## 1577 0.001696167 -0.02196061 0.02535295 -0.03448376 0.03787609
##
## Call:
## arima(x = dlogprice_training, order = c(2, 0, 2), include.mean = FALSE)
##
## Coefficients:
## ar1 ar2 ma1 ma2
## 0.3377 -0.2877 -0.2326 0.2194
## s.e. 0.3461 0.2394 0.3492 0.2575
##
## sigma^2 estimated as 0.0003405: log likelihood = 4058.48, aic = -8106.96
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.00144696 0.01845375 0.0127338 99.9059 145.5905 0.7352664
## ACF1
## Training set -0.006089293
##
## Forecast method: ARIMA(2,0,2) with zero mean
##
## Model Information:
##
## Call:
## arima(x = dlogprice_training, order = c(2, 0, 2), include.mean = FALSE)
##
## Coefficients:
## ar1 ar2 ma1 ma2
## 0.3377 -0.2877 -0.2326 0.2194
## s.e. 0.3461 0.2394 0.3492 0.2575
##
## sigma^2 estimated as 0.0003405: log likelihood = 4058.48, aic = -8106.96
##
## Error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.00144696 0.01845375 0.0127338 99.9059 145.5905 0.7352664
## ACF1
## Training set -0.006089293
##
## Forecasts:
## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## 1578 -0.0009322146 -0.02458164 0.02271721 -0.03710089 0.03523646
##
## Call:
## arima(x = dlogprice_training, order = c(2, 0, 2), include.mean = FALSE)
##
## Coefficients:
## ar1 ar2 ma1 ma2
## 0.3346 -0.2934 -0.2296 0.2263
## s.e. 0.3423 0.2399 0.3454 0.2574
##
## sigma^2 estimated as 0.0003406: log likelihood = 4060.98, aic = -8111.97
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.001457578 0.01845453 0.01273821 99.9986 145.3978 0.7354571
## ACF1
## Training set -0.006026158
##
## Forecast method: ARIMA(2,0,2) with zero mean
##
## Model Information:
##
## Call:
## arima(x = dlogprice_training, order = c(2, 0, 2), include.mean = FALSE)
##
## Coefficients:
## ar1 ar2 ma1 ma2
## 0.3346 -0.2934 -0.2296 0.2263
## s.e. 0.3423 0.2399 0.3454 0.2574
##
## sigma^2 estimated as 0.0003406: log likelihood = 4060.98, aic = -8111.97
##
## Error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.001457578 0.01845453 0.01273821 99.9986 145.3978 0.7354571
## ACF1
## Training set -0.006026158
##
## Forecasts:
## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## 1579 0.00143986 -0.02221057 0.02509029 -0.03473036 0.03761008
##
## Call:
## arima(x = dlogprice_training, order = c(2, 0, 2), include.mean = FALSE)
##
## Coefficients:
## ar1 ar2 ma1 ma2
## 0.3304 -0.2884 -0.2250 0.2215
## s.e. 0.3445 0.2383 0.3477 0.2553
##
## sigma^2 estimated as 0.0003404: log likelihood = 4063.85, aic = -8117.71
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.001463069 0.01845107 0.01273766 100.0329 145.4919 0.7357423
## ACF1
## Training set -0.006139675
##
## Forecast method: ARIMA(2,0,2) with zero mean
##
## Model Information:
##
## Call:
## arima(x = dlogprice_training, order = c(2, 0, 2), include.mean = FALSE)
##
## Coefficients:
## ar1 ar2 ma1 ma2
## 0.3304 -0.2884 -0.2250 0.2215
## s.e. 0.3445 0.2383 0.3477 0.2553
##
## sigma^2 estimated as 0.0003404: log likelihood = 4063.85, aic = -8117.71
##
## Error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.001463069 0.01845107 0.01273766 100.0329 145.4919 0.7357423
## ACF1
## Training set -0.006139675
##
## Forecasts:
## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## 1580 0.000665633 -0.02298036 0.02431163 -0.0354978 0.03682907
##
## Call:
## arima(x = dlogprice_training, order = c(2, 0, 2), include.mean = FALSE)
##
## Coefficients:
## ar1 ar2 ma1 ma2
## 0.3367 -0.2879 -0.2317 0.2199
## s.e. 0.3432 0.2416 0.3463 0.2593
##
## sigma^2 estimated as 0.0003405: log likelihood = 4066.26, aic = -8122.51
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.001450613 0.01845306 0.01274294 99.90457 145.3775 0.7356007
## ACF1
## Training set -0.006094123
##
## Forecast method: ARIMA(2,0,2) with zero mean
##
## Model Information:
##
## Call:
## arima(x = dlogprice_training, order = c(2, 0, 2), include.mean = FALSE)
##
## Coefficients:
## ar1 ar2 ma1 ma2
## 0.3367 -0.2879 -0.2317 0.2199
## s.e. 0.3432 0.2416 0.3463 0.2593
##
## sigma^2 estimated as 0.0003405: log likelihood = 4066.26, aic = -8122.51
##
## Error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.001450613 0.01845306 0.01274294 99.90457 145.3775 0.7356007
## ACF1
## Training set -0.006094123
##
## Forecasts:
## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## 1581 -0.003239006 -0.02688756 0.02040954 -0.03940634 0.03292833
##
## Call:
## arima(x = dlogprice_training, order = c(2, 0, 2), include.mean = FALSE)
##
## Coefficients:
## ar1 ar2 ma1 ma2
## 0.3356 -0.2880 -0.2312 0.2204
## s.e. 0.3435 0.2434 0.3466 0.2610
##
## sigma^2 estimated as 0.0003404: log likelihood = 4069.15, aic = -8128.31
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.001456722 0.01844931 0.01274169 99.94739 145.1253 0.7352278
## ACF1
## Training set -0.006036143
##
## Forecast method: ARIMA(2,0,2) with zero mean
##
## Model Information:
##
## Call:
## arima(x = dlogprice_training, order = c(2, 0, 2), include.mean = FALSE)
##
## Coefficients:
## ar1 ar2 ma1 ma2
## 0.3356 -0.2880 -0.2312 0.2204
## s.e. 0.3435 0.2434 0.3466 0.2610
##
## sigma^2 estimated as 0.0003404: log likelihood = 4069.15, aic = -8128.31
##
## Error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.001456722 0.01844931 0.01274169 99.94739 145.1253 0.7352278
## ACF1
## Training set -0.006036143
##
## Forecasts:
## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## 1582 0.001326101 -0.02231764 0.02496984 -0.03483388 0.03748608
##
## Call:
## arima(x = dlogprice_training, order = c(2, 0, 2), include.mean = FALSE)
##
## Coefficients:
## ar1 ar2 ma1 ma2
## 0.3360 -0.2880 -0.2315 0.2204
## s.e. 0.3434 0.2433 0.3465 0.2609
##
## sigma^2 estimated as 0.0003402: log likelihood = 4072.23, aic = -8134.45
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.001455659 0.01844348 0.0127338 99.87112 145.0523 0.7350574
## ACF1
## Training set -0.006060042
##
## Forecast method: ARIMA(2,0,2) with zero mean
##
## Model Information:
##
## Call:
## arima(x = dlogprice_training, order = c(2, 0, 2), include.mean = FALSE)
##
## Coefficients:
## ar1 ar2 ma1 ma2
## 0.3360 -0.2880 -0.2315 0.2204
## s.e. 0.3434 0.2433 0.3465 0.2609
##
## sigma^2 estimated as 0.0003402: log likelihood = 4072.23, aic = -8134.45
##
## Error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.001455659 0.01844348 0.0127338 99.87112 145.0523 0.7350574
## ACF1
## Training set -0.006060042
##
## Forecasts:
## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## 1583 0.0006025782 -0.02303369 0.02423884 -0.03554597 0.03675113
##
## Call:
## arima(x = dlogprice_training, order = c(2, 0, 2), include.mean = FALSE)
##
## Coefficients:
## ar1 ar2 ma1 ma2
## 0.3361 -0.2884 -0.2316 0.2208
## s.e. 0.3429 0.2431 0.3460 0.2606
##
## sigma^2 estimated as 0.0003399: log likelihood = 4075.3, aic = -8140.6
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.001455232 0.01843766 0.01272623 99.84296 144.9957 0.7350777
## ACF1
## Training set -0.006052488
##
## Forecast method: ARIMA(2,0,2) with zero mean
##
## Model Information:
##
## Call:
## arima(x = dlogprice_training, order = c(2, 0, 2), include.mean = FALSE)
##
## Coefficients:
## ar1 ar2 ma1 ma2
## 0.3361 -0.2884 -0.2316 0.2208
## s.e. 0.3429 0.2431 0.3460 0.2606
##
## sigma^2 estimated as 0.0003399: log likelihood = 4075.3, aic = -8140.6
##
## Error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.001455232 0.01843766 0.01272623 99.84296 144.9957 0.7350777
## ACF1
## Training set -0.006052488
##
## Forecasts:
## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## 1584 -8.278886e-05 -0.0237116 0.02354602 -0.03621994 0.03605436
##
## Call:
## arima(x = dlogprice_training, order = c(2, 0, 2), include.mean = FALSE)
##
## Coefficients:
## ar1 ar2 ma1 ma2
## 0.3391 -0.3079 -0.2351 0.2407
## s.e. 0.3335 0.2453 0.3365 0.2635
##
## sigma^2 estimated as 0.0003398: log likelihood = 4078.12, aic = -8146.24
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.001446991 0.01843478 0.01272644 99.77431 144.8273 0.7351627
## ACF1
## Training set -0.005590442
##
## Forecast method: ARIMA(2,0,2) with zero mean
##
## Model Information:
##
## Call:
## arima(x = dlogprice_training, order = c(2, 0, 2), include.mean = FALSE)
##
## Coefficients:
## ar1 ar2 ma1 ma2
## 0.3391 -0.3079 -0.2351 0.2407
## s.e. 0.3335 0.2453 0.3365 0.2635
##
## sigma^2 estimated as 0.0003398: log likelihood = 4078.12, aic = -8146.24
##
## Error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.001446991 0.01843478 0.01272644 99.77431 144.8273 0.7351627
## ACF1
## Training set -0.005590442
##
## Forecasts:
## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## 1585 -0.001647186 -0.02527231 0.02197794 -0.03777869 0.03448432
##
## Call:
## arima(x = dlogprice_training, order = c(2, 0, 2), include.mean = FALSE)
##
## Coefficients:
## ar1 ar2 ma1 ma2
## 0.3318 -0.2894 -0.2273 0.2222
## s.e. 0.3433 0.2414 0.3464 0.2588
##
## sigma^2 estimated as 0.0003396: log likelihood = 4081.19, aic = -8152.38
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.001443504 0.01842905 0.01271972 99.80101 144.8539 0.7349824
## ACF1
## Training set -0.005990802
##
## Forecast method: ARIMA(2,0,2) with zero mean
##
## Model Information:
##
## Call:
## arima(x = dlogprice_training, order = c(2, 0, 2), include.mean = FALSE)
##
## Coefficients:
## ar1 ar2 ma1 ma2
## 0.3318 -0.2894 -0.2273 0.2222
## s.e. 0.3433 0.2414 0.3464 0.2588
##
## sigma^2 estimated as 0.0003396: log likelihood = 4081.19, aic = -8152.38
##
## Error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.001443504 0.01842905 0.01271972 99.80101 144.8539 0.7349824
## ACF1
## Training set -0.005990802
##
## Forecasts:
## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## 1586 0.0001538969 -0.02346389 0.02377168 -0.03596638 0.03627418
##
## Call:
## arima(x = dlogprice_training, order = c(2, 0, 2), include.mean = FALSE)
##
## Coefficients:
## ar1 ar2 ma1 ma2
## 0.3321 -0.2894 -0.2276 0.2222
## s.e. 0.3432 0.2414 0.3463 0.2588
##
## sigma^2 estimated as 0.0003394: log likelihood = 4084.26, aic = -8158.53
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.001442042 0.01842326 0.01271225 99.81836 144.8456 0.7349352
## ACF1
## Training set -0.005976674
##
## Forecast method: ARIMA(2,0,2) with zero mean
##
## Model Information:
##
## Call:
## arima(x = dlogprice_training, order = c(2, 0, 2), include.mean = FALSE)
##
## Coefficients:
## ar1 ar2 ma1 ma2
## 0.3321 -0.2894 -0.2276 0.2222
## s.e. 0.3432 0.2414 0.3463 0.2588
##
## sigma^2 estimated as 0.0003394: log likelihood = 4084.26, aic = -8158.53
##
## Error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.001442042 0.01842326 0.01271225 99.81836 144.8456 0.7349352
## ACF1
## Training set -0.005976674
##
## Forecasts:
## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## 1587 0.0005697673 -0.02304058 0.02418012 -0.03553915 0.03667868
##
## Call:
## arima(x = dlogprice_training, order = c(2, 0, 2), include.mean = FALSE)
##
## Coefficients:
## ar1 ar2 ma1 ma2
## 0.3320 -0.2901 -0.2275 0.2229
## s.e. 0.3426 0.2413 0.3457 0.2587
##
## sigma^2 estimated as 0.0003392: log likelihood = 4087.33, aic = -8164.67
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.001442131 0.01841749 0.01270522 99.79971 144.7931 0.7349159
## ACF1
## Training set -0.005965339
##
## Forecast method: ARIMA(2,0,2) with zero mean
##
## Model Information:
##
## Call:
## arima(x = dlogprice_training, order = c(2, 0, 2), include.mean = FALSE)
##
## Coefficients:
## ar1 ar2 ma1 ma2
## 0.3320 -0.2901 -0.2275 0.2229
## s.e. 0.3426 0.2413 0.3457 0.2587
##
## sigma^2 estimated as 0.0003392: log likelihood = 4087.33, aic = -8164.67
##
## Error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.001442131 0.01841749 0.01270522 99.79971 144.7931 0.7349159
## ACF1
## Training set -0.005965339
##
## Forecasts:
## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## 1588 0.0003666669 -0.0232363 0.02396963 -0.03573095 0.03646428
##
## Call:
## arima(x = dlogprice_training, order = c(2, 0, 2), include.mean = FALSE)
##
## Coefficients:
## ar1 ar2 ma1 ma2
## 0.3338 -0.2906 -0.2293 0.2232
## s.e. 0.3415 0.2417 0.3446 0.2592
##
## sigma^2 estimated as 0.000339: log likelihood = 4090.35, aic = -8170.7
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.001445418 0.01841239 0.01270128 99.79533 144.8023 0.7350268
## ACF1
## Training set -0.005969215
##
## Forecast method: ARIMA(2,0,2) with zero mean
##
## Model Information:
##
## Call:
## arima(x = dlogprice_training, order = c(2, 0, 2), include.mean = FALSE)
##
## Coefficients:
## ar1 ar2 ma1 ma2
## 0.3338 -0.2906 -0.2293 0.2232
## s.e. 0.3415 0.2417 0.3446 0.2592
##
## sigma^2 estimated as 0.000339: log likelihood = 4090.35, aic = -8170.7
##
## Error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.001445418 0.01841239 0.01270128 99.79533 144.8023 0.7350268
## ACF1
## Training set -0.005969215
##
## Forecasts:
## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## 1589 0.0005193535 -0.02307707 0.02411578 -0.03556826 0.03660697
##
## Call:
## arima(x = dlogprice_training, order = c(2, 0, 2), include.mean = FALSE)
##
## Coefficients:
## ar1 ar2 ma1 ma2
## 0.3342 -0.2857 -0.2294 0.2181
## s.e. 0.3466 0.2397 0.3497 0.2574
##
## sigma^2 estimated as 0.000339: log likelihood = 4092.87, aic = -8175.74
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.001456194 0.01841305 0.01270563 99.83401 144.9301 0.735389
## ACF1
## Training set -0.006135375
##
## Forecast method: ARIMA(2,0,2) with zero mean
##
## Model Information:
##
## Call:
## arima(x = dlogprice_training, order = c(2, 0, 2), include.mean = FALSE)
##
## Coefficients:
## ar1 ar2 ma1 ma2
## 0.3342 -0.2857 -0.2294 0.2181
## s.e. 0.3466 0.2397 0.3497 0.2574
##
## sigma^2 estimated as 0.000339: log likelihood = 4092.87, aic = -8175.74
##
## Error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.001456194 0.01841305 0.01270563 99.83401 144.9301 0.735389
## ACF1
## Training set -0.006135375
##
## Forecasts:
## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## 1590 0.001673608 -0.02192366 0.02527088 -0.0344153 0.03776251
##
## Call:
## arima(x = dlogprice_training, order = c(2, 0, 2), include.mean = FALSE)
##
## Coefficients:
## ar1 ar2 ma1 ma2
## 0.3316 -0.2846 -0.2265 0.2173
## s.e. 0.3487 0.2384 0.3518 0.2561
##
## sigma^2 estimated as 0.0003389: log likelihood = 4095.89, aic = -8181.77
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.001458507 0.01840793 0.01270164 99.85441 144.9542 0.7352984
## ACF1
## Training set -0.006231067
##
## Forecast method: ARIMA(2,0,2) with zero mean
##
## Model Information:
##
## Call:
## arima(x = dlogprice_training, order = c(2, 0, 2), include.mean = FALSE)
##
## Coefficients:
## ar1 ar2 ma1 ma2
## 0.3316 -0.2846 -0.2265 0.2173
## s.e. 0.3487 0.2384 0.3518 0.2561
##
## sigma^2 estimated as 0.0003389: log likelihood = 4095.89, aic = -8181.77
##
## Error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.001458507 0.01840793 0.01270164 99.85441 144.9542 0.7352984
## ACF1
## Training set -0.006231067
##
## Forecasts:
## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## 1591 -0.000244168 -0.02383487 0.02334654 -0.03632304 0.0358347
##
## Call:
## arima(x = dlogprice_training, order = c(2, 0, 2), include.mean = FALSE)
##
## Coefficients:
## ar1 ar2 ma1 ma2
## 0.3310 -0.2845 -0.2258 0.2174
## s.e. 0.3491 0.2385 0.3522 0.2561
##
## sigma^2 estimated as 0.0003386: log likelihood = 4098.95, aic = -8187.9
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.001459303 0.01840231 0.01269566 99.88108 144.9314 0.7352808
## ACF1
## Training set -0.00625168
##
## Forecast method: ARIMA(2,0,2) with zero mean
##
## Model Information:
##
## Call:
## arima(x = dlogprice_training, order = c(2, 0, 2), include.mean = FALSE)
##
## Coefficients:
## ar1 ar2 ma1 ma2
## 0.3310 -0.2845 -0.2258 0.2174
## s.e. 0.3491 0.2385 0.3522 0.2561
##
## sigma^2 estimated as 0.0003386: log likelihood = 4098.95, aic = -8187.9
##
## Error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.001459303 0.01840231 0.01269566 99.88108 144.9314 0.7352808
## ACF1
## Training set -0.00625168
##
## Forecasts:
## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## 1592 -0.0006447112 -0.02422822 0.0229388 -0.03671258 0.03542315
##
## Call:
## arima(x = dlogprice_training, order = c(2, 0, 2), include.mean = FALSE)
##
## Coefficients:
## ar1 ar2 ma1 ma2
## 0.3271 -0.2817 -0.2219 0.2151
## s.e. 0.3517 0.2381 0.3547 0.2555
##
## sigma^2 estimated as 0.0003385: log likelihood = 4101.91, aic = -8193.83
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.001463084 0.01839781 0.01269307 99.92036 144.862 0.7354581
## ACF1
## Training set -0.006309886
##
## Forecast method: ARIMA(2,0,2) with zero mean
##
## Model Information:
##
## Call:
## arima(x = dlogprice_training, order = c(2, 0, 2), include.mean = FALSE)
##
## Coefficients:
## ar1 ar2 ma1 ma2
## 0.3271 -0.2817 -0.2219 0.2151
## s.e. 0.3517 0.2381 0.3547 0.2555
##
## sigma^2 estimated as 0.0003385: log likelihood = 4101.91, aic = -8193.83
##
## Error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.001463084 0.01839781 0.01269307 99.92036 144.862 0.7354581
## ACF1
## Training set -0.006309886
##
## Forecasts:
## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## 1593 0.0005544239 -0.02302332 0.02413217 -0.03550462 0.03661347
##
## Call:
## arima(x = dlogprice_training, order = c(2, 0, 2), include.mean = FALSE)
##
## Coefficients:
## ar1 ar2 ma1 ma2
## 0.3390 -0.2852 -0.2343 0.2174
## s.e. 0.3497 0.2416 0.3527 0.2601
##
## sigma^2 estimated as 0.0003387: log likelihood = 4103.93, aic = -8197.86
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.001447186 0.01840428 0.01270193 99.87159 144.8502 0.7355165
## ACF1
## Training set -0.006193087
##
## Forecast method: ARIMA(2,0,2) with zero mean
##
## Model Information:
##
## Call:
## arima(x = dlogprice_training, order = c(2, 0, 2), include.mean = FALSE)
##
## Coefficients:
## ar1 ar2 ma1 ma2
## 0.3390 -0.2852 -0.2343 0.2174
## s.e. 0.3497 0.2416 0.3527 0.2601
##
## sigma^2 estimated as 0.0003387: log likelihood = 4103.93, aic = -8197.86
##
## Error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.001447186 0.01840428 0.01270193 99.87159 144.8502 0.7355165
## ACF1
## Training set -0.006193087
##
## Forecasts:
## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## 1594 -0.003024574 -0.02661061 0.02056147 -0.03909631 0.03304716
##
## Call:
## arima(x = dlogprice_training, order = c(2, 0, 2), include.mean = FALSE)
##
## Coefficients:
## ar1 ar2 ma1 ma2
## 0.3376 -0.2741 -0.2313 0.2054
## s.e. 0.3603 0.2370 0.3631 0.2563
##
## sigma^2 estimated as 0.000339: log likelihood = 4105.91, aic = -8201.82
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.001427669 0.01841117 0.01271151 99.90071 145.5109 0.7364265
## ACF1
## Training set -0.006239062
##
## Forecast method: ARIMA(2,0,2) with zero mean
##
## Model Information:
##
## Call:
## arima(x = dlogprice_training, order = c(2, 0, 2), include.mean = FALSE)
##
## Coefficients:
## ar1 ar2 ma1 ma2
## 0.3376 -0.2741 -0.2313 0.2054
## s.e. 0.3603 0.2370 0.3631 0.2563
##
## sigma^2 estimated as 0.000339: log likelihood = 4105.91, aic = -8201.82
##
## Error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.001427669 0.01841117 0.01271151 99.90071 145.5109 0.7364265
## ACF1
## Training set -0.006239062
##
## Forecasts:
## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## 1595 -0.002238523 -0.02583339 0.02135634 -0.03832375 0.03384671
##
## Call:
## arima(x = dlogprice_training, order = c(2, 0, 2), include.mean = FALSE)
##
## Coefficients:
## ar1 ar2 ma1 ma2
## 0.3374 -0.2741 -0.2310 0.2054
## s.e. 0.3602 0.2369 0.3631 0.2562
##
## sigma^2 estimated as 0.0003388: log likelihood = 4108.99, aic = -8207.97
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.001426645 0.0184054 0.01270363 99.84314 145.4239 0.7356856
## ACF1
## Training set -0.006166497
##
## Forecast method: ARIMA(2,0,2) with zero mean
##
## Model Information:
##
## Call:
## arima(x = dlogprice_training, order = c(2, 0, 2), include.mean = FALSE)
##
## Coefficients:
## ar1 ar2 ma1 ma2
## 0.3374 -0.2741 -0.2310 0.2054
## s.e. 0.3602 0.2369 0.3631 0.2562
##
## sigma^2 estimated as 0.0003388: log likelihood = 4108.99, aic = -8207.97
##
## Error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.001426645 0.0184054 0.01270363 99.84314 145.4239 0.7356856
## ACF1
## Training set -0.006166497
##
## Forecasts:
## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## 1596 0.001942324 -0.02164514 0.02552979 -0.0341316 0.03801624
plot(arima.forecast)
#Visualizing and validating model results
realreturn = realreturn[-1]
forecastreturn = xts(forecastreturn, index(realreturn))
plot(realreturn,type = 'l',main = 'Actual Returns(black) vs Forecast Returns(red)')
lines(forecastreturn ,lwd = 2, col = 'red')
comparison = merge(realreturn,forecastreturn)
comparison
## realreturn Forecasted
## 2021-02-19 -0.0086476863 1.080032e-04
## 2021-02-22 -0.0229021775 -1.098070e-03
## 2021-02-23 -0.0251087778 -2.095314e-03
## 2021-02-24 0.0102203454 -1.280204e-03
## 2021-02-25 -0.0251489405 2.801507e-03
## 2021-02-26 -0.0036355145 -2.263543e-03
## 2021-03-01 0.0199821275 6.354664e-05
## 2021-03-02 -0.0035148555 2.829266e-03
## 2021-03-03 -0.0302944238 -1.050912e-03
## 2021-03-04 -0.0166894380 -3.725869e-03
## 2021-03-05 -0.0221173374 -3.001465e-04
## 2021-03-08 0.0241117035 -4.101203e-04
## 2021-03-09 0.0180782914 3.969307e-03
## 2021-03-10 0.0082516863 1.264959e-03
## 2021-03-11 0.0173012842 -9.802688e-04
## 2021-03-12 -0.0122153301 7.475144e-04
## 2021-03-15 -0.0044129116 -2.091077e-03
## 2021-03-16 0.0143323920 -2.890042e-04
## 2021-03-17 -0.0018353609 2.170725e-03
## 2021-03-18 -0.0148381193 -5.831170e-04
## 2021-03-19 -0.0027708160 -2.013532e-03
## 2021-03-22 0.0142909962 3.770919e-04
## 2021-03-23 0.0196741603 2.188571e-03
## 2021-03-24 -0.0115051852 1.508138e-03
## 2021-03-25 -0.0156839989 -2.684923e-03
## 2021-03-26 -0.0136381822 -1.808345e-03
## 2021-03-29 0.0107728556 -1.571180e-04
## 2021-03-30 0.0018309596 2.477296e-03
## 2021-03-31 0.0093496949 7.571212e-05
## 2021-04-01 0.0171741415 2.887444e-04
## 2021-04-05 0.0145586351 1.185614e-03
## 2021-04-06 0.0174973417 5.314097e-04
## 2021-04-07 0.0020525333 6.872809e-04
## 2021-04-08 0.0209810553 -9.602613e-04
## 2021-04-09 -0.0009421502 1.696167e-03
## 2021-04-12 0.0187504087 -9.322146e-04
## 2021-04-13 0.0131857351 1.439860e-03
## 2021-04-14 -0.0207188956 6.656330e-04
## 2021-04-15 0.0077868018 -3.239006e-03
## 2021-04-16 0.0010704555 1.326101e-03
## 2021-04-19 0.0013609276 6.025782e-04
## 2021-04-20 -0.0132293284 -8.278886e-05
## 2021-04-21 -0.0036828715 -1.647186e-03
## 2021-04-22 -0.0007146147 1.538969e-04
## 2021-04-23 0.0021331634 5.697673e-04
## 2021-04-26 0.0067596143 3.666669e-04
## 2021-04-27 0.0199354404 5.193535e-04
## 2021-04-28 0.0079115026 1.673608e-03
## 2021-04-29 0.0029154540 -2.441680e-04
## 2021-04-30 0.0079739458 -6.447112e-04
## 2021-05-03 -0.0262772755 5.544239e-04
## 2021-05-04 -0.0302724828 -3.024574e-03
## 2021-05-05 -0.0023773553 -2.238523e-03
## 2021-05-06 -0.0052706865 1.942324e-03
comparison$Accuracy = sign(comparison$realreturn)==sign(comparison$Forecasted)
Accuracy_percentage = sum(comparison$Accuracy == 1)*100/length(comparison$Accuracy)
Accuracy_percentage
## [1] 61.11111
